Ada and Paul received their scores on three tests. On the first test, Ada's score was 10 points higher than...

1. Translate the problem requirements: We need to find how many points higher Paul scored than Ada on the third test. We know Ada scored 10 points higher on test 1, 4 points higher on test 2, but Paul's overall average was 3 points higher than Ada's average.
2. Set up variables for the scoring relationship: Define Paul's scores as our base variables and express Ada's scores relative to Paul's, which keeps the algebra simpler.
3. Apply the average relationship constraint: Use the fact that Paul's average exceeded Ada's average by 3 points to create an equation linking all test scores.
4. Solve for the third test difference: Manipulate the equation to isolate the difference between Paul's and Ada's third test scores.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday language:

• Ada scored 10 points higher than Paul on test 1
• Ada scored 4 points higher than Paul on test 2
• Paul's average across all three tests was 3 points higher than Ada's average
• We need to find: How many points higher did Paul score than Ada on test 3?

This might seem contradictory at first - Ada did better on the first two tests, but Paul had a higher overall average. This means Paul must have done significantly better than Ada on the third test to overcome his deficits from the first two tests.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Set up variables for the scoring relationship

Let's use Paul's scores as our starting point since the question asks about Paul's performance relative to Ada's:

• Let Paul's score on test 1 = \(\mathrm{P_1}\)
• Let Paul's score on test 2 = \(\mathrm{P_2}\)
• Let Paul's score on test 3 = \(\mathrm{P_3}\)

Now we can express Ada's scores in terms of Paul's scores:

• Ada's score on test 1 = \(\mathrm{P_1} + 10\) (since she scored 10 higher)
• Ada's score on test 2 = \(\mathrm{P_2} + 4\) (since she scored 4 higher)
• Ada's score on test 3 = \(\mathrm{P_3} - x\) (where x is what we're looking for - how many points Paul scored higher than Ada)

3. Apply the average relationship constraint

Now let's think about the averages. Paul's average is his total points divided by 3:

Paul's average = \(\frac{\mathrm{P_1} + \mathrm{P_2} + \mathrm{P_3}}{3}\)

Ada's average is her total points divided by 3:

Ada's average = \(\frac{(\mathrm{P_1} + 10) + (\mathrm{P_2} + 4) + (\mathrm{P_3} - x)}{3}\)

This simplifies to: \(\frac{\mathrm{P_1} + \mathrm{P_2} + \mathrm{P_3} + 14 - x}{3}\)

We know that Paul's average is 3 points higher than Ada's average, so:

\(\frac{\mathrm{P_1} + \mathrm{P_2} + \mathrm{P_3}}{3} = \frac{\mathrm{P_1} + \mathrm{P_2} + \mathrm{P_3} + 14 - x}{3} + 3\)

4. Solve for the third test difference

Let's solve this step by step. First, let's multiply everything by 3 to eliminate fractions:

\(\mathrm{P_1} + \mathrm{P_2} + \mathrm{P_3} = (\mathrm{P_1} + \mathrm{P_2} + \mathrm{P_3} + 14 - x) + 9\)

Now we can subtract \((\mathrm{P_1} + \mathrm{P_2} + \mathrm{P_3})\) from both sides:

\(0 = 14 - x + 9\)

\(0 = 23 - x\)

\(x = 23\)

Let's verify this makes sense: Ada was ahead by \(10 + 4 = 14\) points after two tests. For Paul to have an average that's 3 points higher across three tests, he needs to overcome this 14-point deficit plus gain an additional 9 points (since \(3 \text{ points} \times 3 \text{ tests} = 9\) total points). Therefore, Paul needs to outscore Ada by \(14 + 9 = 23\) points on the third test.

Process Skill: MANIPULATE - Algebraic manipulation to isolate the unknown variable

4. Final Answer

Paul's score on the third test was 23 points higher than Ada's score.

The answer is (D) 23.

Common Faltering Points

Errors while devising the approach

• Misinterpreting the direction of score differences: Students often get confused about who scored higher than whom, especially when the problem states "Ada's score was 10 points higher than Paul's" but then reveals "Paul's average was 3 points higher than Ada's average." This apparent contradiction can lead students to set up their variables incorrectly or flip the relationships.
• Incorrectly defining the third test variable: When setting up the equation for Ada's third test score, students might define it as "\(\mathrm{P_3} + x\)" (thinking Ada scored higher) instead of "\(\mathrm{P_3} - x\)" (where x represents how much Paul scored higher), because they assume the pattern from the first two tests continues.
• Misunderstanding the average relationship constraint: Students may incorrectly interpret "Paul's average was 3 points higher" and try to add 3 to individual test scores rather than understanding this refers to the overall average across all three tests, leading to incorrect equation setup.

Errors while executing the approach

• Sign errors when manipulating the average equation: When solving the equation \(\frac{\mathrm{P_1} + \mathrm{P_2} + \mathrm{P_3}}{3} = \frac{\mathrm{P_1} + \mathrm{P_2} + \mathrm{P_3} + 14 - x}{3} + 3\), students often make sign errors, particularly when moving terms across the equals sign or when handling the "-x" term in the equation.
• Arithmetic mistakes in the final calculation: Students may correctly set up the equation but make computational errors when solving \(0 = 23 - x\), possibly getting \(x = -23\) instead of \(x = 23\), or making errors when combining the deficit (14 points) with the additional advantage needed (9 points).

Errors while selecting the answer

• Selecting the wrong direction of the difference: Even after correctly calculating that the difference is 23 points, students might second-guess themselves due to the counterintuitive nature of the problem (Ada was ahead on first two tests but Paul has higher average) and incorrectly conclude that Ada scored 23 points higher than Paul on the third test.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient values for Paul's test scores

Let's assign specific values to Paul's scores that will make our calculations clean. We'll choose:

• Paul's score on test 1: 80 points
• Paul's score on test 2: 70 points
• Paul's score on test 3: we'll call this \(\mathrm{P_3}\) (to be determined)

Step 2: Calculate Ada's scores on tests 1 and 2

From the given information:

• Ada's score on test 1 = Paul's score + 10 = \(80 + 10 = 90\) points
• Ada's score on test 2 = Paul's score + 4 = \(70 + 4 = 74\) points
• Ada's score on test 3 = we'll call this \(\mathrm{A_3}\) (to be determined)

Step 3: Set up the average relationship

Paul's average = \(\frac{80 + 70 + \mathrm{P_3}}{3} = \frac{150 + \mathrm{P_3}}{3}\)

Ada's average = \(\frac{90 + 74 + \mathrm{A_3}}{3} = \frac{164 + \mathrm{A_3}}{3}\)

Given that Paul's average is 3 points higher than Ada's average:

\(\frac{150 + \mathrm{P_3}}{3} = \frac{164 + \mathrm{A_3}}{3} + 3\)

Step 4: Solve for the relationship between \(\mathrm{P_3}\) and \(\mathrm{A_3}\)

Multiply both sides by 3:

\(150 + \mathrm{P_3} = 164 + \mathrm{A_3} + 9\)

\(150 + \mathrm{P_3} = 173 + \mathrm{A_3}\)

\(\mathrm{P_3} = 173 + \mathrm{A_3} - 150\)

\(\mathrm{P_3} = 23 + \mathrm{A_3}\)

Step 5: Find how many points higher Paul scored than Ada on test 3

\(\mathrm{P_3} - \mathrm{A_3} = (23 + \mathrm{A_3}) - \mathrm{A_3} = 23\)

Therefore, Paul's score on the third test was 23 points higher than Ada's score on the third test.

Verification: Let's check with \(\mathrm{A_3} = 77\), so \(\mathrm{P_3} = 100\)

• Paul's scores: 80, 70, 100 → Average = \(\frac{250}{3} \approx 83.33\)
• Ada's scores: 90, 74, 77 → Average = \(\frac{241}{3} \approx 80.33\)
• Difference in averages: \(83.33 - 80.33 = 3\) ✓